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= Corso di INFORMATICA = '''Corso di Laurea in Matematica''' ---- [[Anchor(TOP)]] [[BR]] [[Anchor(TOC)]] [[TableOfContents]] ---- [[Anchor(docente)]] == Docente == '''Francesco Calimeri''' [http://www.mat.unical.it/calimeri link alla home page] [[BR]] Dipartimento di Matematica, Università della Calabria, 87036 Rende (CS). [[BR]] Ufficio: cubo 30B, 2°piano [[BR]] Telefono: +39 0984 496478 [[BR]] Fax: +39 0984 496410 [[BR]] Email: calimeri[AT]mat.unical.it [[BR]] [[BR]] [#TOP Top of Page] ---- [[Anchor(materiale)]] == Testi e Materiale Didattico == === Testi consigliati === ''Testi'' * Roberto Bruni, Andrea Corradini, Vincenzo Gervasi, ''Programmazione in Java'', APOGEO, 2009. * Harvey M. Deitel, Paul J. Deitel, ''Java. Fondamenti di programmazione'', APOGEO, 2005. * Cay Horstmann, ''Concetti di informatica e fondamenti di Java'', APOGEO, 2007 ''Varie e sparse'' * D. C. Dennett, D. R. Hofstadter, "L'io della mente. Fantasie e riflessioni sul se' e sull'anima", Adelphi Ed., 1997. [http://en.wikipedia.org/wiki/The_Mind's_I link a wikipedia] * D. R. Hofstadter, "Goedel, Escher, Bach: un'eterna ghirlanda brillante", Adelphi Ed., 1990. [http://en.wikipedia.org/wiki/Goedel_Escher_Bach link a wikipedia] === Materiale Didattico, Links, Varie ed Eventuali === ''Lucidi'' * - ''Tutorial'' * - ''Links'' * - [#TOP Top of Page] ---- [[Anchor(obiettivi)]] == Obiettivi del Corso == Il corso intende fornire solide basi sulla programmazione dei calcolatori elettronici per la risoluzione di problemi. Esso prevede lo studio di tecniche di programmazione nel linguaggio Java, con uso di semplici strutture dati, ed una introduzione alla programmazione ad oggetti. [[BR]] [#TOP Top of Page] ---- [[Anchor(prerequisiti)]] == Prerequisiti == Sono utili conoscenze di base di matematica, logica, una dimestichezza minima con l'uso del computer. [#TOP Top of Page] ---- [[Anchor(programma)]] == Programma del Corso == * Rappresentazione dell’informazione * Rappresentazione di numeri naturali * Cenni di aritmetica binaria * Rappresentazione di informazione non numerica (caratteri, immagini, ecc.) * Calcolo proposizionale * Architettura del calcolatore * Processore, memoria centrale, memoria di massa, memoria cache, periferiche * Algoritmi * Definizione di algoritmo * Risoluzione algoritmica dei problemi * Linguaggi di Programmazione * Definizione informale di un linguaggio di programmazione * Linguaggi a basso e ad alto livello * Interpreti e Compilatori * Diagrammi di flusso e pseudo-codice * Ambienti integrati di programmazione * Gli ambienti visuali * Programmazione in Java - Primi Passi * Introduzione - Struttura di un programma * Il metodo main * La memoria: concetti fondamentali * Librerie * Operazioni di ingresso/uscita * Concetto di variabile * Inizializzazione e assegnamento * Costanti * Espressioni aritmetiche e booleane * Priorità degli operatori * Ambienti di sviluppo * Compilatore, linker e debugger * Definizione dello spazio di lavoro e di un progetto * L'Editor * Compilazione, esecuzione e debugging di un programma * Tipi Primitivi * Tipi interi, tipi reali, tipo char, tipo bool * Conversioni di tipo e operazioni di cast * Strutture di Controllo * Istruzioni semplici e composte * Definizione di blocco di istruzioni * Visibilità delle variabili * L'istruzione IF * Effettuare confronti * L'istruzione WHILE * L'istruzione FOR * L'istruzione DO-WHILE * L'istruzione SWITCH * L’istruzione BREAK * Istruzioni innestate * Le funzioni in Java – I Metodi * I moduli di programmi in Java * Dichiarazioni e definizioni di metodi * Parametri formali e valore di ritorno * Durata degli identificatori * Regole di visibilità * Concetto di ricorsione * Ricorsione e iterazione * Cenni all’overloading delle funzioni * Array - Riferimenti * Dichiarare e allocare gli array * I riferimenti e i loro parametri * Passare gli array ai metodi * Array multidimensionali * Array di caratteri e stringhe * Concetti fondamentali * Funzioni di libreria * Introduzione alla Programmazione Orientata Agli Oggetti |
#acl EditorsGroup:read,write All:read = Official Problem Suite = == Problem List - Model & Solve and System Track == <<BR>> ||<tablewidth="1024px" tableheight="421px">'''Nr''' ||'''Name''' ||'''Author(s)''' ||'''Type''' ||'''Complexity''' ||'''Domain''' ||'''Problem Description''' ||'''Sample Instances''' ||'''Checker''' || '''ASP-Core-2 encoding''' || ||N01 ||Permutation Pattern Matching ||Martin Lackner, Andreas Pfandler ||Search ||NP ||Combinatorial ||[[https://www.mat.unical.it/aspcomp2013/FinalProblemDescriptions/PermutationPatternMatching|view]] || [[https://www.mat.unical.it/aspcomp2013/files/PPM-samples.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/permutation_pattern_matching-checker.zip|download]] || [[attachment:Permutation-Pattern-Matching-ENCODING.asp|Permutation Pattern Matching]] || ||N02 ||Valves Location Problem ||Andrea Peano ||Optimization ||Beyond NP ||Combinatorial ||[[https://www.mat.unical.it/aspcomp2013/FinalProblemDescriptions/ValvesLocation|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/valves_location_problem-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/valves_location_problem-checker.zip|download]] || [[attachment:Valves-Location-Problem-ENCODING.asp|Valves Location Problem]] || ||N03 ||Rotating Workforce Scheduling ||Francesco Calimeri, Thomas Krennwallner, Nysret Musliu ||Search || ||Scheduling ||[[https://www.mat.unical.it/aspcomp2013/RotatingWorkforceScheduling|view]] || || || - || ||N04 ||Connected Maximum-density Still Life ||Christian Drescher ||Optimization ||Beyond NP ||AI ||[[https://www.mat.unical.it/aspcomp2013/ConnectedMaximumDensityStillLife|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/still_life-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/still_life-checker.zip|download]] || [[attachment:Connected-Maximum-density-Still-Life-ENCODING.asp|Connected Maximum-density Still Life]] || ||N05 ||Graceful Graphs ||Christian Drescher ||Search ||NP ||Graph ||[[https://www.mat.unical.it/aspcomp2013/GracefulGraphs|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/graceful_graphs-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/graceful_graphs-checker.zip|download]] || [[attachment:Graceful-Graphs-ENCODING.asp|Graceful Graphs]] || ||N06 ||Bottle Filling Problem ||Wolfgang Faber ||Search || ||Combinatorial ||[[https://www.mat.unical.it/aspcomp2013/BottleFillingProblem|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/bottle_filling-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/bottle_filling-checker.zip|download]] || [[attachment:Bottle-Filling-Problem-ENCODING.asp|Bottle Filling Problem]] || ||N07 ||Nomystery ||Giovambattista Ianni, Carlos Linares López*, Hootan Nakhost* ||Search || ||Planning ||[[https://www.mat.unical.it/aspcomp2013/Nomystery|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/nomystery-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/nomystery-checker.zip|download]] || [[attachment:Nomystery-ENCODING.asp|Nomystery]] || ||N08 ||Sokoban ||Giovambattista Ianni, Wolfgang Faber*, Carlos Linares López* ||Search || ||Planning ||[[https://www.mat.unical.it/aspcomp2013/Sokoban|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/sokoban-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/sokoban-checker.zip|download]] || [[attachment:Sokoban-ENCODING.asp|Sokoban]] || ||N09 ||Ricochet Robots ||Julius Höfler, Martin Gebser, Philipp Obermeier, Roland Kaminski, Torsten Schaub ||Search || ||Puzzle ||[[https://www.mat.unical.it/aspcomp2013/RicochetRobot|view]] || [[https://www.mat.unical.it/aspcomp2013/files/ricochetSampleInstances.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/ricochet_robot-checker.zip|download]] || [[attachment:Ricochet-Robots-ENCODING.asp|Ricochet Robots]] || ||O10 ||Crossing Minimization ||Carmine Dodaro, Graeme Gange*, Peter Stuckey* ||Optimization ||Beyond NP ||Graph ||[[https://www.mat.unical.it/aspcomp2013/FinalProblemDescriptions/CrossingMinimization|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/crossing_minimization-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/crossing_minimization-checker.zip|download]] || [[attachment:Crossing-Minimization-ENCODING.asp|Crossing Minimization]] || ||O11 ||Reachability ||Carmine Dodaro, Giorgio Terracina* ||Query ||P ||Graph ||[[https://www.mat.unical.it/aspcomp2013/FinalProblemDescriptions/Reachability|view]] || || || [[attachment:Reachability-ENCODING.asp|Reachability]] || ||O12 ||Strategic Companies ||Mario Alviano, Marco Maratea*, Francesco Ricca* ||Query ||Beyond NP ||AI ||[[https://www.mat.unical.it/aspcomp2013/FinalProblemDescriptions/StrategicCompanies|view]] ||[[https://www.mat.unical.it/ aspcomp2013/files/samples/strategic_companies-sample.zip|download]] || || [[attachment:Strategic-Companies-ENCODING.asp|Strategic Companies]] || ||O13 ||Solitaire ||Marcello Balduccini, Yuliya Lierler* ||Search ||NP ||Puzzle ||[[https://www.mat.unical.it/aspcomp2013/Solitaire|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/solitaire_sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/solitaire-checker.zip|download]] || [[attachment:Solitaire-ENCODING.asp|Solitaire]] || ||O14 ||Weighted-Sequence Problem ||Marcello Balduccini, Yuliya Lierlier, Shaden Smith ||Search ||NP ||Database ||[[https://www.mat.unical.it/aspcomp2013/WeightedSequenceProblem|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/weighted_sequence-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/weighted_sequence-checker.zip|download]] || [[attachment:Weighted-Sequence-Problem-ENCODING.asp|Weighted-Sequence Problem]] || ||O15 ||Stable Marriage ||Mario Alviano, Carmine Dodaro, Francesco Ricca ||Search ||P ||Graph ||[[https://www.mat.unical.it/aspcomp2013/StableMarriage|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/stablemarriage-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/stable_marriage-checker.zip|download]] || [[attachment:Stable-Marriage-ENCODING.asp|Stable Marriage]] || ||O16 ||Incremental Scheduling ||Marcello Balduccini, Yuliya Lierler* ||Search ||NP ||Scheduling ||[[https://www.mat.unical.it/aspcomp2013/IncrementalScheduling|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/incremental_scheduling-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/incremental-scheduling-checker.zip|download]] || [[attachment:Incremental-Scheduling-ENCODING.asp|Incremental Scheduling]] || |
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== Problem List - System Track only == <<BR>> ||<tablewidth="1024px" tableheight="421px">'''Nr''' ||'''Name''' ||'''Author(s)''' ||'''Type''' ||'''Complexity''' ||'''Domain''' ||'''ASP-Core-2 Encoding''' ||'''Problem Description''' ||'''Sample Instances''' ||'''Checker''' || '''ASP-Core-2 encoding''' || ||N17 ||Qualitative Spatial Reasoning ||Jason Jingshi Li ||Search ||NP ||Formal logic || ||[[https://www.mat.unical.it/aspcomp2013/FinalProblemDescriptions/QualitativeSpatialTemporalReasoning|view]] || ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/qualitative_spatial_reasoning-checker.zip|download]]|| [[attachment:Qualitative-Spatial-Reasoning-ENCODING.asp|Qualitative Spatial Reasoning]] || ||N18 ||Chemical Classification ||Despoina Magka ||Search ||P ||Natural Sciences || ||[[https://www.mat.unical.it/aspcomp2013/ChemicalClassification|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/chemical_classification-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/chemical_classification-checker.zip|download]]|| [[attachment:Chemical-Classification-ENCODING.asp|Chemical Classification]] || ||N19 ||Abstract Dialectical Frameworks Well-founded Model ||Stefan Ellmauthaler, Johannes Wallner ||Optimization || ||Formal logic || ||[[https://www.mat.unical.it/aspcomp2013/AbstractDialecticalFrameworks|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/abstract_dialectical_frameworks-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/abstract_dialectical_frameworks-checker.zip|download]]|| [[attachment:Abstract-Dialectical-Frameworks-Well-founded-Model-ENCODING.asp|Abstract Dialectical Frameworks Well-founded Model]] || ||N20 ||Visit-all ||Giovambattista Ianni,Nir Lipovetzky*, Carlos Linares López* ||Search || ||Planning || ||[[https://www.mat.unical.it/aspcomp2013/VisitAll|view]] || [[https://www.mat.unical.it/aspcomp2013/files/Visitall-samples.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/visit_all-checker.zip|download]]|| [[attachment:Visit-all-ENCODING.asp|Visit-all]] || ||N21 ||Complex Optimization of Answer Sets ||Martin Gebser, Roland Kaminski, Torsten Schaub ||Search || ||Synthetic || ||[[https://www.mat.unical.it/aspcomp2013/ComplexOptimizationAnswerSets|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/complex_optimization-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/complex_optimization-checker.zip|download]]|| [[attachment:Complex-Optimization-of-Answer-Sets-ENCODING.asp|Complex Optimization of Answer Sets]] || ||O22 ||Knight Tour with Holes ||Francesco Calimeri, Neng-Fa Zhou* ||Search ||NP ||Puzzle || ||[[https://www.mat.unical.it/aspcomp2013/KnightTour|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/knight_tour-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/knight_tour-checker.zip|download]]|| [[attachment:Knight-Tour-with-Holes-ENCODING.asp|Knight Tour with Holes]] || ||O23 ||Maximal Clique Problem ||Guenther Charwat, Martin Kronegger, Johan Wittocx* ||Optimization ||Beyond NP ||Graph || ||[[https://www.mat.unical.it/aspcomp2013/MaximalClique|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/maximal_clique-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/maximal_clique-checker.zip|download]]|| [[attachment:Maximal-Clique-Problem-ENCODING.asp|Maximal Clique Problem]] || ||O24 ||Labyrinth ||Carmine Dodaro, Giovambattista Ianni, Martin Gebser* ||Search ||NP ||Puzzle || ||[[https://www.mat.unical.it/aspcomp2013/Labyrinth|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/labyrinth-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/labyrinth-checker.zip|download]]|| [[attachment:Labyrinth-ENCODING.asp|Labyrinth]] || ||O25 ||Minimal Diagnosis ||Marcello Balduccini, Martin Gebser* ||Search ||Beyond NP ||Diagnosis || ||[[https://www.mat.unical.it/aspcomp2013/MinimalDiagnosis|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/minimal_diagnosis-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/minimal_diagnosis-checker.zip|download]]|| [[attachment:Minimal-Diagnosis-ENCODING.asp|Minimal Diagnosis]] || ||O26 ||Hanoi Tower ||Gayathri Namasivayam, Miroslaw Truszczynski, Shaden Smith, Alex Westlund ||Search ||NP ||AI || ||[[https://www.mat.unical.it/aspcomp2013/HanoiTower|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/hanoi_tower-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/hanoi_tower-checker.zip|download]]|| [[attachment:Hanoi-Tower-ENCODING.asp|Hanoi Tower]] || ||O27 ||Graph Colouring ||Johannes Wallner, Marcello Balduccini*, Yuliya Lierler* ||Search ||NP ||Graph || ||[[https://www.mat.unical.it/aspcomp2013/GraphColouring|view]] ||[[https://www.mat.unical.it/aspcomp2013/files/samples/graph_colouring-sample.zip|download]] ||[[https://www.mat.unical.it/aspcomp2013/files/checkers/graph_colouring-checker.zip|download]]|| [[attachment:Graph-Colouring-ENCODING.asp|Graph Colouring]] || |
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[#TOP Top of Page] ---- Torna alla pagina ["didattica"] |
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Torna alla home page di [http://www.mat.unical.it/calimeri Francesco Calimeri] ---- |
== Legend == Columns in the table below are to be read as following: * '''Nr'''. ID of the benchmark. The prefixes ''N'' resp. ''O'' denote a newly added resp. a 2011 benchmark. * '''Author(s)'''. Names of authors. A ''^*^'' denotes that they authored the respective package in previous years. * '''Type'''. One of ''Search'', ''Optimization'', ''Query''. See the problem [[BenchmarkProblems|classification]] page. * '''Final Problem Description'''. Problem specifications in their official form, with latest notes and updates. * '''Sample Instances'''. A link to sample instances of the given problem. |
Official Problem Suite
Problem List - Model & Solve and System Track
Nr |
Name |
Author(s) |
Type |
Complexity |
Domain |
Problem Description |
Sample Instances |
Checker |
ASP-Core-2 encoding |
N01 |
Permutation Pattern Matching |
Martin Lackner, Andreas Pfandler |
Search |
NP |
Combinatorial |
||||
N02 |
Valves Location Problem |
Andrea Peano |
Optimization |
Beyond NP |
Combinatorial |
||||
N03 |
Rotating Workforce Scheduling |
Francesco Calimeri, Thomas Krennwallner, Nysret Musliu |
Search |
|
Scheduling |
|
|
- |
|
N04 |
Connected Maximum-density Still Life |
Christian Drescher |
Optimization |
Beyond NP |
AI |
||||
N05 |
Graceful Graphs |
Christian Drescher |
Search |
NP |
Graph |
||||
N06 |
Bottle Filling Problem |
Wolfgang Faber |
Search |
|
Combinatorial |
||||
N07 |
Nomystery |
Giovambattista Ianni, Carlos Linares López*, Hootan Nakhost* |
Search |
|
Planning |
||||
N08 |
Sokoban |
Giovambattista Ianni, Wolfgang Faber*, Carlos Linares López* |
Search |
|
Planning |
||||
N09 |
Ricochet Robots |
Julius Höfler, Martin Gebser, Philipp Obermeier, Roland Kaminski, Torsten Schaub |
Search |
|
Puzzle |
||||
O10 |
Crossing Minimization |
Carmine Dodaro, Graeme Gange*, Peter Stuckey* |
Optimization |
Beyond NP |
Graph |
||||
O11 |
Reachability |
Carmine Dodaro, Giorgio Terracina* |
Query |
P |
Graph |
|
|
||
O12 |
Strategic Companies |
Mario Alviano, Marco Maratea*, Francesco Ricca* |
Query |
Beyond NP |
AI |
|
|||
O13 |
Solitaire |
Marcello Balduccini, Yuliya Lierler* |
Search |
NP |
Puzzle |
||||
O14 |
Weighted-Sequence Problem |
Marcello Balduccini, Yuliya Lierlier, Shaden Smith |
Search |
NP |
Database |
||||
O15 |
Stable Marriage |
Mario Alviano, Carmine Dodaro, Francesco Ricca |
Search |
P |
Graph |
||||
O16 |
Incremental Scheduling |
Marcello Balduccini, Yuliya Lierler* |
Search |
NP |
Scheduling |
Problem List - System Track only
Nr |
Name |
Author(s) |
Type |
Complexity |
Domain |
ASP-Core-2 Encoding |
Problem Description |
Sample Instances |
Checker |
ASP-Core-2 encoding |
N17 |
Qualitative Spatial Reasoning |
Jason Jingshi Li |
Search |
NP |
Formal logic |
|
|
|||
N18 |
Chemical Classification |
Despoina Magka |
Search |
P |
Natural Sciences |
|
||||
N19 |
Abstract Dialectical Frameworks Well-founded Model |
Stefan Ellmauthaler, Johannes Wallner |
Optimization |
|
Formal logic |
|
||||
N20 |
Visit-all |
Giovambattista Ianni,Nir Lipovetzky*, Carlos Linares López* |
Search |
|
Planning |
|
||||
N21 |
Complex Optimization of Answer Sets |
Martin Gebser, Roland Kaminski, Torsten Schaub |
Search |
|
Synthetic |
|
||||
O22 |
Knight Tour with Holes |
Francesco Calimeri, Neng-Fa Zhou* |
Search |
NP |
Puzzle |
|
||||
O23 |
Maximal Clique Problem |
Guenther Charwat, Martin Kronegger, Johan Wittocx* |
Optimization |
Beyond NP |
Graph |
|
||||
O24 |
Labyrinth |
Carmine Dodaro, Giovambattista Ianni, Martin Gebser* |
Search |
NP |
Puzzle |
|
||||
O25 |
Minimal Diagnosis |
Marcello Balduccini, Martin Gebser* |
Search |
Beyond NP |
Diagnosis |
|
||||
O26 |
Hanoi Tower |
Gayathri Namasivayam, Miroslaw Truszczynski, Shaden Smith, Alex Westlund |
Search |
NP |
AI |
|
||||
O27 |
Graph Colouring |
Johannes Wallner, Marcello Balduccini*, Yuliya Lierler* |
Search |
NP |
Graph |
|
Legend
Columns in the table below are to be read as following:
Nr. ID of the benchmark. The prefixes N resp. O denote a newly added resp. a 2011 benchmark.
Author(s). Names of authors. A * denotes that they authored the respective package in previous years.
Type. One of Search, Optimization, Query. See the problem classification page.
Final Problem Description. Problem specifications in their official form, with latest notes and updates.
Sample Instances. A link to sample instances of the given problem.